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When simulating incompressible flow around airfoil using vortex methods of computational hydrodynamics, it is necessary to solve the boundary integral equation with respect to the intensity of the vortex sheet at the airfoil surface line. For its approximate solution, the Galerkin method with a piecewise-constant and piecewise-linear representation of the solution on the airfoil surface line can be applied. The coefficients of the system of linear equations, arising from the procedure of discretization of the integral equation, e xpress the influence of the vortex sheet located on the panels of the airfoil surface line on other panels; the components of the right-hand side vector express the influence of point vortices that simulate a vortex wake in the flow domain. For the case of the airfoil surface line approximation with a polyline consisting of rectilinear panels, exact analytical expressions are obtained for the coefficients of the matrix and the right-hand side vector; for the case of taking into account the curvature of the panels, an approach for their approximate calculation is proposed and all the necessary formulae are derived.

A model of the nonlinear interaction between a pressure perturbation traveling at a constant velocity and an incompressible boundary layer is constructed when its near-wall part is described by the “inviscid boundary layer” equations. A steady-state solution is managed to obtain in the finite form under the assumption that it exists in a moving coordinate system. It is shown that the boundary layer can easily overcome pressure perturbations whose amplitude is not higher than the dynamic pressure calculated from the velocity of the pressure perturbation. At the higher pressure perturbation amplitudes a vortex sheet sheds from the body surface to the boundary layer. The vortex sheet represents an unstable surface of the tangential discontinuity which separates the regions of the direct and reverse separation flows. In the case of an arbitrary shape of the pressure perturbation the surface of the tangential discontinuity sheds from the body surface at a finite angle with the formation of a stagnation point. An example of the pressure perturbation in which the vortex sheet sheds from the body surface along the tangent is constructed.

During the mixing of viscous incompressible flows with different velocities, in the vicinity of a trailing edge an interaction region with a three-layer structure is formed, similar to that in the case of symmetric shedding with equal velocities. The boundary layers developing on the upper and lower sides of the airfoil form a viscous mixing layer, or vortex sheet, which separates the flows downstream of the trailing edge. The boundary value problem corresponding to the flow in the viscous sublayer in the vicinity of the trailing edge of a flat plate is solved for high Reynolds numbers using an efficient numerical method for solving the equations of asymptotic interaction theory.

A method is developed for simulating the motion and deformation of a tangential discontinuity with stability characteristics similar to the real ones. As distinct from the discrete vortex method, which forms the basis of the method proposed, the motion of a continuous vortex sheet of finite thickness is considered. The equations of motion are derived on the basis of an analysis of the physical reasons for the stability of this sheet with respect to small-scale perturbations.

The paper presents the results of an experimental study of the vortex street behind a circular cylinder oscillating across a water stream at Reynolds numbers less than the critical value at which a vortex street is formed behind a stationary cylinder. The asymptotic dependence of the nondimensional vortex separation frequency (the Strouhal number) on the main relevant parameters is obtained. The value of a constant coefficient appearing in this relationship has been found experimentally.

A model for describing the vertically averaged vortex motions of an incompressible viscous fluid with an arbitrary vertical structure of the bottom Ekman boundary layer is proposed. An approach similar to that adopted in [1] is used: the second moments of the deviations from the average velocities required in order to close the vorticity equation are calculated by means of the Ekman solution for gradient flows, which makes it possible to take the integral bottom boundary layer effect into account. As a result, these terms lead to a specific form of nonlinear friction with a coefficient that depends on the vorticity of the average flow. In the particular case of a constant vertical turbulent transfer coefficient the inaccuracies of the model described in [1] can be eliminated. The generalized vorticity equation obtained has solutions of the vorticity spot type with a uniform internal vorticity distribution, which can be effectively investigated by means of appropriate algorithms [2]. The mechanism of entrainment at the vorticity front is illustrated with reference to the example of the evolution of vorticity spots. An exact solution of the problem of the evolution of an elliptic vortex (generalized Kirchhoff vortex), which in the case of fairly strong anticyclonic vorticity degenerates first into a line segment (vortex sheet) and then into a point vortex, is constructed. Equations describing the dynamics of an elliptic vorticity spot in an external field with a linear dependence of the velocity on the horizontal coordinates and generalizing the classical Chaplygin-Kida model [3, 4] are constructed.

The research carried out in [1–8] is developed by considering the self-similar problem of the unsteady separated flow over a plate expanding from a point with the constant velocity D of a plane-parallel stream of ideal fluid with velocity V∞. At infinity the flow is uniform, steady and normal to the surface of the plate. A wide range of values of the parameter α=V∞/D is investigated. On the value of α there depends, in particular, the direction of shedding of the vortex sheets (VS) which, in accordance with the Joukowsky-Chaplygin condition, occur in separated flow over a plate. A comparison is made with the results obtained when the sheets are replaced by vortex filaments (VF). In accordance with [9] the choice of the intensity of the VF ensures, like the introduction of VS, the finiteness of the flow velocity at the edges of the plate. Within the framework of the unsteady analogy and the law of plane sections the problem of the flow over a delta wing at an angle of attack reduces to the unsteady flow over an expanding plate investigated. In addition to [3, 9], this question was also examined in [10–15]. In [11–15] and in [3] the analysis is based on VS and in [9, 10] on VF. Special attention is paid to the topology of the flow, in particular, to the structure of the so-called conical streamlines and their points of convergence and divergence (this was done in [3] for a special, nonlinear law of expansion of the plate and a variable free-stream velocity). The results obtained for the models with VS and VF are compared over a broad range of values of α, not only with respect to the integral characteristics, as in [12], but also with respect to the flow patterns.

The method of discrete vortices, described in [1, 2], is used to make a detailed analysis of special features of separated flow over an infinite flat plate at angles of attack 0

The study discusses problems in the numerical modeling of axisymmetric flows of an incompressible fluid induced by a continuous vortex region or vortex sheet by means of a discrete system of vortex rings.

An asymptotic solution is constructed to the problem of the flow of a viscous incompressible fluid in the neighborhood of the axis of a vortex sheet generated by flow separation from sharp edges of a delta wing of small aspect ratio at large values of the Reynolds number and small angles of attack.

The method of integral equations is generalized to calculate steady flow past wings with an arbitrary shape in plan with subsonic leading and trailing edges. The determination of the velocity potential in the leading part of the wing, where there is no influence of the vortex sheet, is reduced to the solution of a two-dimensional integral equation of the second kind. The trailing part, which is subject to the influence of the vortex sheet, is divided into a number of subregions, in which the calculation of the acceleration potential reduces to the solution of one-dimensional equations of the type of Fredholm equations of the second kind and to quadrature. The unique solvability of the obtained integral equations is investigated; it is shown that they can be solved by successive approximation. As an example, the solution to the problem of flow past a flat delta-shaped wing is found and compared with the exact solution to the problem found by the method of conic flows [4, 6].

An effective means of controlling wing leading-edge stall at high angles of attack is deflection of the nose in order to assure shock-free entrance of the stream. A numerical method of computing the angles of nose deflection and the aerodynamic characteristics of a thin wing of arbitrary planform for a shock-free entrance of the steady ideal incompressible fluid stream is elucidated in this paper on the basis of nonlinear wing theory [1]. The problem is solved by the method of discrete vortices. In the computations, the wing and its wake, replaced by a vortex sheet, are modeled by a system of discrete vortices which are nonlinear segments with constant circulation along the length. The angles of deflection of the nose and the aerodynamic characteristics of the wing, including shunting of the free vortices shed from the side and trailing edges, are determined during the computation. Examples of an electronic digital computer are presented.